Re: st: Multinomial logit-type probability in log likelihood forML model

[Bob Hammond <robert.g.hammond@vanderbilt.edu>]

Hey Maarten,

Thanks for your reply.  The objective in the example you sent is to 
estimate a multinomial logit (MNL) model, but my objective is 
different.  I want to estimate a binary choice model where the 
probability of Y=1 is governed by a log likelihood function that 
contains a MNL-type probability.  I can't simply use a MNL model because 
I do not observe the outcome of the MN process.  If the MN process is 
described by Binomial(n, j, q), then I observe n, but not j, preventing 
me from using Stata's built-in MNL.  My simplification of the log 
likelihood function made this less clear than it should have been; sorry 
for the confusion.  My approach is to model q as a function of observed 
covariates, allowing me to use the distribution of the unknown j.

After working more on this, I wonder if I can use the following 
(maintaining my simplification from the initial email that ignores the 
fact that j is unobserved):

gen a = .
...
egen double `tempvar1' = sum(exp(`theta'))
replace a = 1 + `tempvar1'
replace `lnf' = (Binomial(n,j,exp(`theta')/a)  ///
   - Binomial(n,j+1,exp(`theta')/a) * f(x) if $ML_y1 == 1

That is, can I simply sum up the `theta' for the entire data set into a 
new variable, then use this variable in the "replace `lnf'" statement?  
Thanks,

Bob

> I posted a -ml- program that does a multinomial logit on statalist some
> time ago:
> http://www.stata.com/statalist/archive/2007-05/msg00449.html
>
> Hope this helps,
> Maarten
>
>
> --- Bob Hammond <robert.g.hammond@vanderbilt.edu> wrote:
>
> > All,
> >
> > I am having trouble coding an ML program for my log likelihood
> > function.  I'll simplify several aspects of the log likelihood that
> > seem
> > unrelated to my question.  Define `lnf' for an individual observation
> > as
> > follows:
> >
> > replace `lnf' = (Binomial(n,j,q) - Binomial(n,j+1,q)) * f(x)  if
> > $ML_y1 == 1
> >
> > In words, the probability that Y is 1 is the probability of observing
> >
> > exactly j successes out of n (where q is the probability of a success
> > on
> > an individual trial) times some function f(x).  The probability q
> > takes
> > a multinomial logit form:
> >
> > q_i = exp(X_i * `theta') / (1 + sum(exp(X_h * `theta')))
> >
> > where the sum goes from h=1, ..., J, so it sums the product of the
> > covariate vector X times the parameter vector `theta' for all
> > observations in the data set.
> >
> > It seems that I need some way to construct the summation in the
> > denominator of q first, but my confusion is that this denominator
> > contains the `theta' parameter vector.  Basically, for every
> > observation
> > in the data set, I need to construct a scalar that multiplies the 1 x
> > k
> > covariate vector X by the k x 1 parameter vector `theta' that needs
> > to
> > be estimated.  Then I need to sum these J scalars up (call this a,
> > which
> > is a function of `theta').  If I could do that, then, after
> > constructing
> > a, I would write:
> >
> > replace `lnf' = (Binomial(n,j, exp(`theta') / (1 + a) - Binomial(n,j,
> >
> > exp(`theta') / (1 + a)) * f(x)
> >             if $ML_y1 == 1
> >
> > I don't know how to construct "a" because it contains the parameter
> > vector `theta'.  I've tried taking a look at the ado file for mlogit,
> >
> > but didn't find the answer.  Thanks in advance,
> >
> > Bob
--
------------------------------------------------------------------------
Bob Hammond
Department of Economics
Vanderbilt University
http://people.vanderbilt.edu/~robert.g.hammond/
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